2Warm Up 1. What is the name of the point where the angle bisectors of a triangle intersect? Find the midpoint of the segment with the given endpoints. 2. (1, 6) and (3, 0) 3. (7, 2) and (3, 8) 4. Write an equation of the line containing the points (3, 1) and (2, 10) in point-slope form. incenter(1, 3)(5, 3)y 1 9(x 3) 3Your Goal Today isApply properties of medians of a triangle. Apply properties of altitudes of a triangle. 4Vocabularymedian of a triangle centroid of a triangle altitude of a triangle orthocenter of a triangle 5A median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side.Every triangle has three medians, and the medians are concurrent. 6The point of concurrency of the medians of a triangle is the centroid of the triangle . The centroid is always inside the triangle. The centroid is also called the center of gravity because it is the point where a triangular region will balance. 7Example 1A Using the Centroid to Find Segment LengthsIn ?LMN, RL 21 and SQ 4. Find LS.Centroid Thm.Substitute 21 for RL.LS 14Simplify. 8Example 1B Using the Centroid to Find Segment LengthsIn ?LMN, RL 21 and SQ 4. Find NQ.Centroid Thm.NS SQ NQSeg. Add. Post.Substitute 4 for SQ.12 NQMultiply both sides by 3. 9Your Turn! Example 1a In ?JKL, ZW 7, and LX 8.1. Find KW.Centroid Thm.Substitute 7 for ZW.Multiply both sides by 3.KW 21 10Your Turn! Example 1b In ?JKL, ZW 7, and LX 8.1. Find LZ.Centroid Thm.Substitute 8.1 for LX.Simplify.LZ 5.4 11Example 2 Problem-Solving ApplicationA sculptor is shaping a triangular piece of iron that will balance on the point of a cone. At what coordinates will the triangular region balance? 12Example 2 ContinuedThe answer will be the coordinates of the centroid of the triangle. The important information is the location of the vertices, A(6, 6), B(10, 7), and C(8, 2). The centroid of the triangle is the point of intersection of the three medians. So write the equations for two medians and find their point of intersection. 13Example 2 Continued 14Example 2 ContinuedLook Back 15An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side. Every triangle has three altitudes. An altitude can be inside, outside, or on the triangle. 16(No Transcript) 17(No Transcript) 18Example 3 Finding the OrthocenterFind the orthocenter of ?XYZ with vertices X(3, 2), Y(3, 6), and Z(7, 1).Step 1 Graph the triangle. 19Example 3 Continued 20Example 3 ContinuedPoint-slope form.Add 6 to both sides. 21Example 3 ContinuedStep 4 Solve the system to find the coordinates of the orthocenter. Substitute 1 for y.Subtract 10 from both sides.6.75 xThe coordinates of the orthocenter are (6.75, 1). 22Your Turn! Example 3 Show that the altitude to JK passes through the orthocenter of ?JKL.4 1 34 4 ?Therefore, this altitude passes through the orthocenter. 23Lesson QuizUse the figure for Items 13. In ?ABC, AE 12, DG 7, and BG 9. Find each length. 1. AG 2. GC 3. GF For Items 4 and 5, use ?MNP with vertices M (4, 2), N (6, 2) , and P (2, 10). Find the coordinates of each point. 4. the centroid 5. the orthocenter

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